摘要:

Letwbe a string of zeros and ones, and letaw(n) be the function which counts the number of (possibly overlapping) occurrences ofwin the binary expansion ofn. We show that there exists an effectively computable rational functionbw) such that∑n⩾0log2(bw(n))Xaw(n)=−11−XBy settingX= − 1 and exponentiating, we recover previous results and also obtain some new ones; for example,∏n⩾1(2n2n+1)(−1)a0(n)=22Our work is a generalization of previous results of D. Woods, D. Robbins, H. Cohen, M. Mendes France, and the authors.

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.193

出版商:Oxford University Press    

年代:2016

数据来源: WILEY

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.205

出版商:Oxford University Press    

年代:2016

数据来源: WILEY

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.219

出版商:Oxford University Press    

年代:2016

数据来源: WILEY

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.231

出版商:Oxford University Press    

年代:2016

数据来源: WILEY

摘要:

Denote bykt(G) the number of complete subgraphs of ordertin the graphG. Letc(t)= min {kt(G) + kt(G); |G| = n}(nt)−1whereG¯denotes the complement ofGand |G| the number of vertices. A well‐known conjecture of Erdős, related to Ramsey's theorem, is that limn→∞ct(n) = 21−(½). This latter number is the proportion of monochromaticKt's in a random colouring ofKn. We present counterexamples to this conjecture and discuss properties of the extremal graphs.

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.246

出版商:Oxford University Press    

年代:2016

数据来源: WILEY

摘要:

We study the rate at which a positive superharmonic function u can tend to zero at a boundary pointz0of a plane domainG. In particular, ifGis a quasidisk, and α>0 is given, we show that the condition that lim infu(z)/dist (z, ∂G)1/α>0 asz0inGfor any suchuis related to the condition that the conformal mapfof the unit disk ontoGwithaf(1) =zois Hölder continuous with exponent α at the point 1. This leads us to consider the problem of finding the best exponent α for whichfis Hölder continuous. The answer depends on how we characterize quasidisks or quasicircles. In this connection we give a negative answer to a question of Näkki and Palka.

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.256

出版商:Oxford University Press    

年代:2016

数据来源: WILEY

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.271

出版商:Oxford University Press    

年代:2016

数据来源: WILEY

摘要:

We define theJohn constantγ(D) of a domainD⊂Cto be sup{α ⩾ 1:1 ⩽ |f′(z)|⩽ α inDimplies thatfis univalent in D), and consider the case in whichDis the upper half‐planeHor a Jordan domain with sufficiently smooth boundary. A functionf0is called an extremal function for such aDif 1 ⩾|f′0(z)| ⩾ γ(D) onDandf0(a)=fo(b) for two distinct pointsa,b∈δD. A simple compactness argument shows that there exist extremal functions forH. LetBN(D, α) = {f:f′ =eαh, where Re{h} is the harmonic measure ofNthe union ofNarcs on ∂D}. It is shown thatf0is an extremal function forD, thenf∈BN(D, In γ(D)) for someN. As a corollary we deduce that for any such D there existsKsuch that γ(D) = sup{ex: allfinBK(D, α) are univalent inD}; in particular this holds whenDis the unit disk.

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.285

出版商:Oxford University Press    

年代:2016

数据来源: WILEY

摘要:

We prove inequalities of the type ∥|A|B−B|A|∥p⩽kp∥AB−BA∥pfor various classes of bounded operators on a Hilbert space, and various values ofpin the range 1

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.299

出版商:Oxford University Press    

年代:2016

数据来源: WILEY

ISSN:0024-6107    

DOI:10.1112/jlms/s2-39.2.309

出版商:Oxford University Press    

年代:2016

数据来源: WILEY